Best Known (58, 72, s)-Nets in Base 8
(58, 72, 4706)-Net over F8 — Constructive and digital
Digital (58, 72, 4706)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 25)-net over F8, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 4 and N(F) ≥ 25, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
- digital (47, 61, 4681)-net over F8, using
- net defined by OOA [i] based on linear OOA(861, 4681, F8, 14, 14) (dual of [(4681, 14), 65473, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(861, 32767, F8, 14) (dual of [32767, 32706, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(861, 32767, F8, 14) (dual of [32767, 32706, 15]-code), using
- net defined by OOA [i] based on linear OOA(861, 4681, F8, 14, 14) (dual of [(4681, 14), 65473, 15]-NRT-code), using
- digital (4, 11, 25)-net over F8, using
(58, 72, 18725)-Net in Base 8 — Constructive
(58, 72, 18725)-net in base 8, using
- base change [i] based on digital (40, 54, 18725)-net over F16, using
- net defined by OOA [i] based on linear OOA(1654, 18725, F16, 14, 14) (dual of [(18725, 14), 262096, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(1654, 131075, F16, 14) (dual of [131075, 131021, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1654, 131076, F16, 14) (dual of [131076, 131022, 15]-code), using
- trace code [i] based on linear OA(25627, 65538, F256, 14) (dual of [65538, 65511, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(25625, 65536, F256, 13) (dual of [65536, 65511, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(25627, 65538, F256, 14) (dual of [65538, 65511, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1654, 131076, F16, 14) (dual of [131076, 131022, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(1654, 131075, F16, 14) (dual of [131075, 131021, 15]-code), using
- net defined by OOA [i] based on linear OOA(1654, 18725, F16, 14, 14) (dual of [(18725, 14), 262096, 15]-NRT-code), using
(58, 72, 81296)-Net over F8 — Digital
Digital (58, 72, 81296)-net over F8, using
(58, 72, large)-Net in Base 8 — Upper bound on s
There is no (58, 72, large)-net in base 8, because
- 12 times m-reduction [i] would yield (58, 60, large)-net in base 8, but